Differential Topology

Differential topology rectifier design for ambient wireless energy harvesting

2016 IEEE International Conference on RFID Technology and Applications (RFID-TA) ◽

10.1109/rfid-ta.2016.7750756 ◽

2016 ◽

Cited By ~ 3

Author(s):

Miaowang Zeng ◽

Andrey S. Andrenko ◽

Xianluo Liu ◽

Bo Zhu ◽

Zihong Li ◽

...

Keyword(s):

Energy Harvesting ◽

Differential Topology ◽

Wireless Energy Harvesting

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Detection of the homotopy type of an object using differential invariants of an approximating map

Computer Optics ◽

10.18287/2412-6179-2019-43-4-611-617 ◽

2019 ◽

Vol 43 (4) ◽

pp. 611-617

Author(s):

S.V. Kurochkin

Keyword(s):

Neural Network ◽

Data Analysis ◽

Homotopy Type ◽

Morse Theory ◽

Object Representation ◽

Differential Invariants ◽

Topological Data Analysis ◽

Higher Dimension ◽

Differential Topology ◽

Open Questions

A method of topological data analysis is proposed that allows one to find out the homotopy type of the object under study. Unlike mature and widely used methods based on persistent homologies, our method is based on computing differential invariants of some map associated with an approximating map. Differential topology tools and the analogy with the main result in Morse theory are used. The approximating map can be constructed in the usual way using a neural network or otherwise. The method allows one to identify the homotopy type of an object in the plane because the number of circles in the homotopy equivalent object representation as a wedge is expressed through the degree of some map associated with the approximating map. The performance of the algorithm is illustrated by examples from the MNIST database and transforms thereof. Generalizations and open questions relating to a higher-dimension case are discussed.

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Lectures on Differential Topology

10.1090/gsm/218 ◽

2021 ◽

Author(s):

Riccardo Benedetti

Keyword(s):

Differential Topology

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Optimization of Utility Functions in an Admissible Space of Higher Dimension

Handbook of Research on Military, Aeronautical, and Maritime Logistics and Operations - Advances in Logistics, Operations, and Management Science ◽

10.4018/978-1-4666-9779-9.ch006 ◽

2016 ◽

pp. 102-113

Author(s):

German Almanza ◽

Victor M. Carrillo ◽

Cely C. Ronquillo

Keyword(s):

Graduate Students ◽

Linear Algebra ◽

Algebraic Topology ◽

Morse Theory ◽

Utility Functions ◽

Higher Dimension ◽

Differential Topology ◽

Several Variables

S. Smale published a paper where announce a theorem which optimize a several utility functions at once (cf. Smale, 1975) using Morse Theory, this is a very abstract subject that require high skills in Differential Topology and Algebraic Topology. Our goal in this paper is announce the same theorems in terms of Calculus of Manifolds and Linear Algebra, those subjects are more reachable to engineers and economists whom are concern with maximizing functions in several variables. Moreover, the elements involved in our theorems are accessible to graduate students, also we putting forward the results we consider economically relevant.

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